Volume 5, Issue 4

We define an antipode of a point p as a farthest point from p . In this paper we describe, on the surface of a convex polyhedron endowed with its intrinsic metric, points admitting at least two antipodes.

Let B be a class of point-line geometries. Given Γ i  ∈  B with subspace i for i  = 1, 2, does any isomorphism Γ 1  −  1  → Γ 2 −  2 extend to a unique isomorphism Γ 1 → Γ 2 ? It is known to be true if B is the class of almost all projective spaces or the class of almost all nondegenerate polar spaces. We show that this is true for the class of almost all strong parapolar spaces, including dual polar spaces. A special case occurs when Γ 1  = Γ 2  = Γ has an embedding into a projective space ℙ( V  ) that is natural in the sense that Aut(Γ ) ≤ PΓL( V  ). Then the question becomes whether ℙ( V  ) is also the natural embedding for Γ − . Our result shows that in most cases the stabilizer Stab Aut(Γ) (Γ − ) is faithful on Γ −  and equals Aut(Γ − ) and so the answer is affirmative. We know that there exist some interesting exceptions. These will be covered in a subsequent paper.

We study different, non-compact groups acting on unitals in compact projective planes in such a way that the geometry can be reconstructed from the group action. These unitals have many ambient automorphisms, but are not induced by polarities. In passing, we construct polarities for a class of eight-dimensional translation planes, and study the corresponding unitals.

We prove that every flock of a finite-dimensional locally compact connected circle plane is homeomorphic to ℝ or 1 and that every flock of a real Miquelian circle plane defines a compact 4-dimensional translation plane. Furthermore we investigate (topological) properties of regulizations. These properties are used to relate the automorphism group of a flock to the automorphism group of the corresponding translation plane.

The paper is a continuation of work initiated by the first two authors in [S. Kuhlmann, M. Marshall, Positivity, sums of squares and the multi-dimensional moment problem. Trans. Amer. Math. Soc. 354 (2002), 4285–4301]. Section 1 is introductory. In Section 2 we prove a basic lemma, Lemma 2.1, and use it to give new proofs of key technical results of Scheiderer in [C. Scheiderer, Sums of squares of regular functions on real algebraic varieties. Trans. Amer. Math. Soc. 352 (2000), 1039–1069] [C. Scheiderer, Sums of squares on real algebraic curves. Math. Z. 245 (2003), 725–760] in the compact case; see Corollaries 2.3, 2.4 and 2.5. Lemma 2.1 is also used in Section 3 where we continue the examination of the case n  = 1 initiated in [S. Kuhlmann, M. Marshall, Positivity, sums of squares and the multi-dimensional moment problem. Trans. Amer. Math. Soc. 354 (2002), 4285–4301], concentrating on the compact case. In Section 4 we prove certain uniform degree bounds for representations in the case n  = 1, which we then use in Section 5 to prove that (‡) holds for basic closed semi-algebraic subsets of cylinders with compact cross-section, provided the generators satisfy certain conditions; see Theorem 5.3 and Corollary 5.5. Theorem 5.3 provides a partial answer to a question raised by Schmüdgen in [K. Schmüdgen, On the moment problem of closed semi-algebraic sets. J. Reine Angew. Math. 558 (2003), 225–234]. We also show that, for basic closed semi-algebraic subsets of cylinders with compact cross-section, the sufficient conditions for (SMP) given in [K. Schmüdgen, On the moment problem of closed semi-algebraic sets. J. Reine Angew. Math. 558 (2003), 225–234] are also necessary; see Corollary 5.2(b). In Section 6 we prove a module variant of the result in [K. Schmüdgen, On the moment problem of closed semi-algebraic sets. J. Reine Angew. Math. 558 (2003), 225–234], in the same spirit as Putinar’s variant [M. Putinar, Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42 (1993), 969–984] of the result in [K. Schmüdgen, The K-moment problem for compact semi-algebraic sets. Math. Ann. 289 (1991), 203–206] in the compact case; see Theorem 6.1. We apply this to basic closed semi-algebraic subsets of cylinders with compact cross-section; see Corollary 6.4. In Section 7 we apply the results from Section 5 to solve two of the open problems listed in [S. Kuhlmann, M. Marshall, Positivity, sums of squares and the multi-dimensional moment problem. Trans. Amer. Math. Soc. 354 (2002), 4285–4301]; see Corollary 7.1 and Corollary 7.4. In Section 8 we consider a number of examples in the plane. In Section 9 we list some open problems.

We show that each of the Artin groups of type B n and D n can be presented as a semidirect product F  ⋊ ℬ n , where F  is a free group and ℬ n is the n -string braid group. We explain how these semidirect product structures arise quite naturally from fibrations, and observe that, in each case, the action of the braid group ℬ n on the free group F is classical. We prove that, for each of the semidirect products, the group of automorphisms which leave invariant the normal subgroup F is small: namely, Out( A ( B n  ), F  ) has order 2, and Out( A ( D n  ), F  ) has order 4 if n is even and 2 if n is odd. It is known that the Artin group of type D n may be viewed as an index 2 subgroup of the n -string braid group over a disk with a degree 2 orbifold point. We show that this orbifold braid group has outer automorphism group of order 2, for all n  ≥ 2.

In this note we address the problem of determining the maximum number of points of intersection of two arithmetically Cohen–Macaulay curves in ℙ 3 . We give a sharp upper bound for the maximum number of points of intersection of two irreducible arithmetically Cohen–Macaulay curves C t and C t–r   in ℙ 3 defined by the maximal minors of a t   × ( t  + 1), resp. ( t  –  r  ) × ( t  –  r  + 1), matrix with linear entries, provided C t–r   has no linear series of degree and dimension n  ≥  t  –  r .

An involutory automorphism of an inversive plane with exactly two fixed points is called an harmonic involution. In this paper we study inversive planes with sufficiently many harmonic involutions satisfying an analog of the theorem about the three reflections.